Dag Prawitz. Hauptsatz for higher order logic. The journal of symbolic logic, Bd. 33 (1968), S. 452–457. - Dag Prawitz. Completeness and Hauptsatz for second order logic. Theoria (Lund), Bd. 33 (1967), S. 246–258. - Moto-o Takahashi. A proof of cut-elimination in simple type-theory. Journal of the Mathematical Society of Japan, Bd. 19 (1967), S. 399–410.

1974 ◽  
Vol 39 (3) ◽  
pp. 607-607
Author(s):  
K. Schütte
Keyword(s):  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
Mathematical Society ◽  
Cut Elimination ◽  
Symbolic Logic ◽  
Simple Type ◽  
Higher Order Logic ◽  
Simple Type Theory ◽  
Second Order Logic

Hauptsatz for higher order logic

1968 ◽  
Vol 33 (3) ◽  
pp. 452-457 ◽  
Author(s):  
Dag Prawitz
Keyword(s):  
Finite Type ◽  
Type Theory ◽  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Simple Type ◽  
Higher Order Logic ◽  
Simple Type Theory ◽  
Second Order Logic

I shall prove in this paper that Gentzen's Hauptsatz is extendible to simple type theory, i.e., to the predicate logic obtained by admitting quantification over predicates of arbitrary finite type and generalizing the second order quantification rules to cover quantifiers of other types. (That Gentzen's Hauptsatz is extendible to this logic has been known as Takeuti's conjecture; see [4].) Gentzen's Hauptsatz has been extended to second order logic in a recent paper by Tait [3]. However, as remarked by Tait, his proof seems not to be extendible to higher orders. The present proof is rather an extension of a different proof of the Hauptsatz for second order logic that I have presented in [1].

On nonstandard models in higher order logic

1984 ◽  
Vol 49 (1) ◽  
pp. 204-219
Author(s):  
Christian Hort ◽  
Horst Osswald
Keyword(s):  
Model Theory ◽  
Type Theory ◽  
Order Logic ◽  
Elementary Embedding ◽  
Simple Type ◽  
Order Model ◽  
Higher Order Logic ◽  
First Order ◽  
Simple Type Theory ◽  
Nonstandard Models

There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn), respectively. If A ≠ ∅ then 〈Aτ〉 is called a model of type logic, if A0 = A and . 〈Aτ〉 is called full if, for every τ = (τ1,…,τn), . The variables for the urelements range over the elements of A and the variables of type (τ1,…, τn) range over those subsets of which are elements of . The theory Th(〈Aτ〉) is the set of all closed formulas in the language which hold in 〈Aτ〉 under natural interpretation of the constants. If 〈Bτ〉 is a model of Th(〈Aτ〉), then there exists a sequence 〈fτ〉 of functions fτ: Aτ → Bτ such that 〈fτ〉 is an elementary embedding from 〈Aτ〉 into 〈Bτ〉. 〈Bτ〉 is called a nonstandard model of 〈Aτ〉, if f0 is not surjective. Otherwise 〈Bτ〉 is called a standard model of 〈Aτ〉.This first concept of model theory in type logic seems to be preferable for applications in model theory, for example in nonstandard analysis, since all nice properties of first order model theory (completeness, compactness, and so on) are preserved.

scholarly journals POLYMORPHISM AND THE OBSTINATE CIRCULARITY OF SECOND ORDER LOGIC: A VICTIMS’ TALE

2018 ◽  
Vol 24 (1) ◽  
pp. 1-52
Author(s):  
PAOLO PISTONE
Keyword(s):  
Type Theory ◽  
Order Type ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Related Notion ◽  
Foundational Research ◽  
The One ◽  
Second Order Logic

AbstractThe investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher-order logic. However, the epistemological significance of such investigations has not received much attention in the contemporary foundational debate.We discuss Girard’s normalization proof for second order type theory or System F and compare it with two faulty consistency arguments: the one given by Frege for the logical system of the Grundgesetze (shown inconsistent by Russell’s paradox) and the one given by Martin-Löf for the intuitionistic type theory with a type of all types (shown inconsistent by Girard’s paradox).The comparison suggests that the question of the circularity of second order logic cannot be reduced to Russell’s and Poincaré’s 1906 “vicious circle” diagnosis. Rather, it reveals a bunch of mathematical and logical ideas hidden behind the hazardous idea of impredicative quantification, constituting a vast (and largely unexplored) domain for foundational research.

A proof of the cut-elimination theorem in simple type theory

1973 ◽  
Vol 38 (2) ◽  
pp. 215-226
Author(s):  
Satoko Titani
Keyword(s):  
Boolean Algebra ◽  
Type Theory ◽  
Arbitrary Number ◽  
Formal System ◽  
Cut Elimination ◽  
Simple Type ◽  
Inductive Definition ◽  
Simple Type Theory ◽  
Maximal Ideals ◽  
Definition Of

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

An untyped higher order logic with Y combinator

2007 ◽  
Vol 72 (4) ◽  
pp. 1385-1404
Author(s):  
James H. Andrews
Keyword(s):  
Model Theory ◽  
Lambda Calculus ◽  
Higher Order ◽  
Order Logic ◽  
Proof System ◽  
Cut Elimination ◽  
Higher Order Logic ◽  
Consistent Model

AbstractWe define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.

scholarly journals Proof normalization modulo

2003 ◽  
Vol 68 (4) ◽  
pp. 1289-1316 ◽  
Author(s):  
Gilles Dowek ◽  
Benjamin Werner
Keyword(s):  
Type Theory ◽  
Cut Elimination ◽  
Simple Type ◽  
First Order ◽  
Simple Type Theory

AbstractWe define a generic notion of cut that applies to many first-order theories. We prove a generic cut elimination theorem showing that the cut elimination property holds for all theories having a so-called pre-model. As a corollary, we retrieve cut elimination for several axiomatic theories, including Church's simple type theory.

scholarly journals A Resolution Calculus for Second-order Logic with Eager Unification

10.29007/zpg2 ◽  
2018 ◽  
Author(s):  
Alexander Leitsch ◽  
Tomer Libal
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Unification Algorithm ◽  
Efficient Search ◽  
First Order ◽  
Unification Algorithms ◽  
Second Order Logic ◽  
Resolution Calculus

The efficiency of the first-order resolution calculus is impaired when lifting it to higher-order logic. The main reason for that is the semi-decidability and infinitary natureof higher-order unification algorithms, which requires the integration of unification within the calculus and results in a non-efficient search for refutations.We present a modification of the constrained resolution calculus (Huet'72) which uses an eager unification algorithm while retaining completeness. Thealgorithm is complete with regard to bounded unification only, which for many cases, does not pose a problem in practice.

Higher‐order Logic Reconsidered

2009 ◽  
Author(s):  
Ignacio Jané
Keyword(s):  
Set Theory ◽  
Logical Consequence ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Consequence Relation ◽  
Fourth Section ◽  
Second Order Logic

This article discusses canonical (i.e., full, or standard) second-order consequence and argues against it being a case of logical consequence. The discussion is divided into three parts. The first part comprises the first three sections. After stating the problem in Section 1, Sections 2 and 3 examine the role that the consequence relation is expected to play in axiomatic theories. This leads to put forward two requirements on logical consequence, which are called “formality” and “noninterference.” It is this last requirement that canonical second-order consequence violates, as the article sets out to substantiate. The fourth section argues that canonical second-order logic is inadequate for axiomatizing set theory, on the grounds that it codes a significant amount of set-theoretical content.

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